Gap statistics for random walks with gamma distributed jumps

Matteo Battilana, Satya N. Majumdar, Gregory Schehr

Published 2017-11-23Version 1

We study the order statistics of a random walk (RW) of $n$ steps whose jumps are distributed according to a gamma distribution $f_p(\eta)\sim |\eta|^p \,e^{-|\eta|}$, where $p$ is a non-negative integer. Our main focus is on the statistics of the gaps $d_{k,n}$ between two successive maxima $d_{k,n}=M_{k,n}-M_{k+1,n}$ where $M_{k,n}$ is the $k$-th maximum of the RW between step 1 and step $n$. In the limit of large $n$, we show that the probability distribution function of the gaps $P_{k,n}(\Delta) = \Pr(d_{k,n} = \Delta)$ reaches a stationary distribution $P_{k,n}(\Delta) \to p_k(\Delta)$. For large $k$, we demonstrate that the typical fluctuations of the gap, for $d_{k,n}= O(1/\sqrt{k})$ (and $n \to \infty$), are described by a non-trivial scaling function that is independent of $k$ and of the jump distribution $f_p(\eta)$, thus corroborating our conjecture about the universality of the regime of typical fluctuations (see G. Schehr, S. N. Majumdar, Phys. Rev. Lett. 108, 040601 (2012)). We also investigate the large fluctuations of the gap, for $d_{k,n} = O(1)$ (and $n \to \infty$), and show that these two regimes of typical and large fluctuations of the gaps match smoothly.